I N S T I T U T E

The Varieties of Democracy Measurement Model: Latent Variable Analysis for Cross-National and Cross-Temporal Expert-Coded Data

Daniel Pemstein, Kyle L. Marquardt, Eitan Tzelgov, Yi-ting Wang, Joshua Krusell and Farhad Miri

Working Paper SERIES 2018:21, 3nd edition

THE VARIETIES OF DEMOCRACY INSTITUTE

April 2018

†

†

i ∈ I

r ∈ R

c ∈ C

t ∈ T = {1, . . . , t}

I i

|R| |I|n = |C|× |T |

tc tc + 1 r

Jr i, c, t

c, t Rct

cr

i

|C|× |T |× |R| y

yctr = zct + ectr

zct c t yctr

r zct ectr r

ectr ∼ F (ectr/σr).

yctr

Jr r

k τr,k−1 < yctr ≤ τr,k τ

τ r = (τr,1, . . . , τr,K−1)

r τr,0 = −∞τr,K = ∞ K

(yctr = k) = (yctr > τr,k−1 ∧ yctr ≤ τr,k)

= (ectr > τr,k−1 − zct ∧ ectr ≤ τr,k − zct)

= F

!τr,k − zct

σr

"− F

!τr,k−1 − zct

σr

"

= F (γr,k − zctβr)− F (γr,k−1 − zctβr) .

βr =1σr

γr,k =τr,kσr

σr r

r

σr i

σr βr

γ

γ τ

t t + 1

F (·)βr

zct

zct

t

t − 1

zc,t ∼ N(zc,t−1, 1)

c

t

γ τ

γ

βr ∼ N (1, 1)

γ

γr,k ∼ N (γcrk , 0.2),

γck ∼ N (γµ

k , 0.2) ,

γµk ∼ U(−6, 6),

γr,k γck

k c

k γµk

γ γµ

γ

z zct ∼N (0, 1)

U(−2, 2) γµk

β γ

γ

z

zct ∼ N (¯yct, 1)

¯yct =yct − ¯y

s,

yct =

#r∈Rct

wctryctr#r∈Rct

wctr,

¯y =

#{c,t}∈CT yct

|C × T | ,

s yct wctr

r

ct

yct ct¯y yct ¯yct

¯yct

N(0, 1)

N(¯yct, 1)

γµ

ct γµk

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F (·)ectr = N (0, σ2

r)

r i σ2r σ2

r < σ2r′ r

z r′

zct

zct ∼ N!actbct

,1

bct

"

act = ¯yct +$

r∈Rct

βryctr bct = 1 +$

r∈Rct

βr.

zct

r ≥ 1.1

yctr

−

−− −

−

−

− − −− −−

−

−−

−

z

βr

ct s

z(s)ct ≤ γµ(s)1 γµ(s)

1 <

z(s)ct ≤ γµ(s)2

p(s)ct,0 = φ(γµ(s)1 − zct

(s))

p(s)ct,1 = φ(γµ(s)2 − zct

(s))− φ(γµ(s)1 − zct

(s))

p(s)ct,2 = φ(γµ(s)3 − zct

(s))− φ(γµ(s)2 − zct

(s))

p(s)ct,3 = 1− φ(γµ(s)3 − zct

(s)).

o(s)ct = 0× p(s)ct,0 + 1× p(s)ct,1 + 2× p(s)ct,2 + 3× p(s)ct,3.

oct

γµ

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